Class 6 Mathematics Notes Chapter 8 (Decimals) – Mathematics Book

Detailed Notes with MCQs of Chapter 8: Decimals. This is a fundamental concept, extending our understanding of numbers, and it frequently appears in various government exams. Pay close attention as we break it down.

Chapter 8: Decimals - Detailed Notes

1. What are Decimals?

• Decimals are a way of representing numbers that are not whole numbers. They represent parts of a whole.
• Think of them as an extension of our place value system to the right of the ones place.
• A decimal point (.) separates the whole number part (on the left) from the fractional part (on the right).
  • Example: In 12.34, '12' is the whole number part, and '34' is the decimal part.

2. Place Value in Decimals

Just like whole numbers have place values (Ones, Tens, Hundreds...), the digits after the decimal point also have place values:

• The first digit after the decimal point is in the Tenths place (representing 1/10).
• The second digit is in the Hundredths place (representing 1/100).
• The third digit is in the Thousandths place (representing 1/1000), and so on.

Place Value Chart Example: Consider the number 235.678

• So, 235.678 = 200 + 30 + 5 + 6/10 + 7/100 + 8/1000

3. Reading Decimals

• Read the whole number part as usual.
• Say "point" for the decimal point.
• Read the digits after the decimal point individually.
  • Example: 12.34 is read as "Twelve point three four".
  • Example: 0.5 is read as "Zero point five" or simply "Point five".

4. Representing Decimals on a Number Line

• The space between any two consecutive whole numbers (e.g., 0 and 1, 1 and 2) can be divided into 10 equal parts. Each part represents a tenth.
• To represent 0.7, divide the space between 0 and 1 into 10 parts and mark the 7th part.
• To represent 1.4, move to 1, then divide the space between 1 and 2 into 10 parts and mark the 4th part.
• Similarly, each tenth can be divided into 10 hundredths.

5. Fractions and Decimals

Decimals are essentially fractions with denominators of 10, 100, 1000, etc.

• Converting Fractions to Decimals:

  • Method 1 (Denominator is 10, 100, 1000...): The number of zeros in the denominator tells you the number of decimal places.
    • 3/10 = 0.3 (One zero, one decimal place)
    • 45/100 = 0.45 (Two zeros, two decimal places)
    • 7/100 = 0.07 (Two zeros, two decimal places - add a leading zero)
    • 123/1000 = 0.123 (Three zeros, three decimal places)
  • Method 2 (Converting Denominator): If the denominator is a factor of 10, 100, 1000..., convert it.
    • 1/2 = (1 × 5) / (2 × 5) = 5/10 = 0.5
    • 3/4 = (3 × 25) / (4 × 25) = 75/100 = 0.75
    • 2/5 = (2 × 2) / (5 × 2) = 4/10 = 0.4
  • Method 3 (Division): Divide the numerator by the denominator (This is a general method).
    • 1/2 = 1 ÷ 2 = 0.5

• Converting Decimals to Fractions:

  • Write the digits after the decimal point as the numerator.
  • The denominator is 1 followed by as many zeros as there are decimal places.
  • Simplify the fraction to its lowest terms.
    • 0.6 = 6/10 = 3/5
    • 0.75 = 75/100 = 3/4
    • 0.05 = 5/100 = 1/20
    • 2.34 = 234/100 = 117/50 (You can also write it as a mixed fraction: 2 34/100 = 2 17/50)

6. Comparing Decimals

To compare two decimals:

• Step 1: Compare the whole number parts. The decimal with the larger whole number part is greater. (e.g., 12.5 > 9.87)
• Step 2: If the whole number parts are equal, compare the tenths digits. The decimal with the larger tenths digit is greater. (e.g., 5.72 > 5.69)
• Step 3: If the tenths digits are also equal, compare the hundredths digits. (e.g., 3.48 > 3.45)
• Step 4: Continue comparing digits place by place from left to right until you find different digits.
• Tip: You can make the number of decimal places equal by adding trailing zeros (zeros at the end after the decimal point). This doesn't change the value.
  • Example: Compare 4.5 and 4.49. Write 4.5 as 4.50. Now compare 4.50 and 4.49. Since 50 > 49, 4.50 > 4.49, so 4.5 > 4.49.
  • Example: Compare 0.07 and 0.1. Write 0.1 as 0.10. Compare 0.07 and 0.10. Since 7 < 10, 0.07 < 0.10, so 0.07 < 0.1.

7. Using Decimals in Real Life

Decimals are commonly used in measurements:

• Money:
  • 100 paise = 1 Rupee (₹)
  • 1 paisa = ₹ 1/100 = ₹ 0.01
  • Example: 65 paise = ₹ 0.65; ₹ 7 and 50 paise = ₹ 7.50
• Length:
  • 10 millimetres (mm) = 1 centimetre (cm) => 1 mm = 1/10 cm = 0.1 cm
  • 100 centimetres (cm) = 1 metre (m) => 1 cm = 1/100 m = 0.01 m
  • 1000 metres (m) = 1 kilometre (km) => 1 m = 1/1000 km = 0.001 km
  • Example: 8 mm = 0.8 cm; 175 cm = 1.75 m; 5 km 30 m = 5.030 km
• Weight (Mass):
  • 1000 grams (g) = 1 kilogram (kg) => 1 g = 1/1000 kg = 0.001 kg
  • Example: 450 g = 0.450 kg = 0.45 kg; 3 kg 5 g = 3.005 kg

8. Addition of Decimals

• Step 1: Write the numbers one below the other such that the decimal points are aligned vertically.

• Step 2: Add placeholder zeros if needed to make the number of decimal places equal.

• Step 3: Add the numbers column by column from right to left, just like whole numbers.

• Step 4: Place the decimal point in the sum directly below the decimal points in the numbers being added.

  Example: Add 21.36 and 37.5
  21.36

  • 37.50 <-- Add placeholder zero

  ---

  58.86

9. Subtraction of Decimals

• Step 1: Write the numbers one below the other, aligning the decimal points vertically (larger number on top).

• Step 2: Add placeholder zeros if needed.

• Step 3: Subtract column by column from right to left, borrowing if necessary, just like whole numbers.

• Step 4: Place the decimal point in the difference directly below the other decimal points.

  Example: Subtract 1.32 from 4.5
  4.50 <-- Add placeholder zero

  • 1.32

  ---

  3.18

Multiple Choice Questions (MCQs)

Here are 10 questions to test your understanding:

1. What is the place value of 7 in the number 29.375?
   a) Tenths
   b) Thousandths
   c) Hundredths
   d) Tens

2. Which of the following represents the fraction 3/5 as a decimal?
   a) 0.3
   b) 0.5
   c) 0.6
   d) 3.5

3. Convert the decimal 0.04 into a fraction in its simplest form.
   a) 4/10
   b) 2/5
   c) 4/1000
   d) 1/25

4. Which of the following decimal numbers is the greatest?
   a) 0.105
   b) 0.501
   c) 0.051
   d) 0.150

5. How can "5 rupees and 8 paise" be written in decimal form?
   a) ₹ 5.80
   b) ₹ 5.08
   c) ₹ 58.00
   d) ₹ 0.58

6. Express 75 mm in cm using decimals.
   a) 0.75 cm
   b) 7.5 cm
   c) 750 cm
   d) 0.075 cm

7. What is the sum of 2.5, 0.05, and 1.11?
   a) 3.66
   b) 2.66
   c) 3.16
   d) 3.61

8. Subtract 3.07 from 8.
   a) 5.07
   b) 4.93
   c) 5.93
   d) 4.30

9. The number 2 + 0 + 5/100 + 9/1000 can be written in decimal form as:
   a) 2.59
   b) 2.059
   c) 2.509
   d) 20.59

10. Which comparison is correct?
    a) 3.30 = 3.03
    b) 1.99 > 1.999
    c) 0.4 = 0.40
    d) 5.64 < 5.46

Answers to MCQs:

1. c) Hundredths
2. c) 0.6 (3/5 = 6/10 = 0.6)
3. d) 1/25 (0.04 = 4/100 = 1/25)
4. b) 0.501
5. b) ₹ 5.08 (8 paise = ₹ 8/100 = ₹ 0.08)
6. b) 7.5 cm (1 mm = 0.1 cm, so 75 mm = 7.5 cm)
7. a) 3.66 (2.50 + 0.05 + 1.11 = 3.66)
8. b) 4.93 (8.00 - 3.07 = 4.93)
9. b) 2.059 (2 + 0 + 0.05 + 0.009 = 2.059)
10. c) 0.4 = 0.40 (Trailing zeros after the decimal point don't change the value)

Revise these notes thoroughly. Understanding decimals is crucial not just for exams but also for everyday calculations involving money and measurements. Let me know if any part needs further clarification.